Fisher's method

In statistics, Fisher's method, developed by and named for Ronald Fisher, is a data fusion or "meta-analysis" (analysis after analysis) technique for combining the results from a variety of independent tests bearing upon the same overall hypothesis (H₀) as if in a single large test.

Fisher's method combines extreme value probabilities, P(results at least as extreme, assuming H₀ true) from each test, called "p-values", into one test statistic (X²) having a chi-square distribution using the formula

<math>X^2_{2k} = -2\sum_{i=1}^k \log_e(p_i).</math>

The p-value for X² itself can then be interpolated from a chi-square table using 2k "degrees of freedom", where k is the number of tests being combined. As in any similar test, H₀ is rejected for small p-values, usually < 0.05.

This figure shows how two p-values ~0.10 (or ~0.04 and ~0.25) combine into one ~0.05.

In the case that the tests are not independent, the null distribution of X² is more complicated. If the correlations between the <math>\log_e(p_i)</math> are known, these can be used to form an approximation.

References

• Fisher, R. A. (1948) "Combining independent tests of significance", American Statistician, vol. 2, issue 5, page 30. (In response to Question 14)

See also

• data fusion
• meta-analysis
• hypothesis test
• p-value
• chi-square distribution
• degrees of freedom (statistics)
• statistical significance
• R. A. Fisher
• deciles